3.39 min read

Linear Maps as Functions

A linear map (or linear transformation) is a function $T: \mathbb{R}^n \to \mathbb{R}^m$ that respects the vector space structure: it preserves addition ( $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ ) and scalar multiplication ( $T(c\mathbf{u}) = cT(\mathbf{u})$ ).

These two conditions together say: $T$ preserves linear combinations. If you know where $T$ sends each basis vector, you know $T$ completely — because every vector is a linear combination of basis vectors.

Examples: rotation, reflection, scaling, projection, shearing. These are all linear maps. Translations (shifting by a fixed vector) are NOT linear (they don't fix the origin).

Formal View

Definition 3.3 — Linear Map

A function

T: \mathbb{R}^n \to \mathbb{R}^m

is a linear map if for all

\mathbf{u}, \mathbf{v} \in \mathbb{R}^n

and

c \in \mathbb{R}

: \begin{enumerate} \item

T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})

\item

T(c\mathbf{u}) = cT(\mathbf{u}) \end{enumerate} Equivalently:

T(c_1\mathbf{u} + c_2\mathbf{v}) = c_1T(\mathbf{u}) + c_2T(\mathbf{v})$.

Why This Matters

Linear maps are the morphisms of linear algebra — the structure-preserving functions between vector spaces.

Computer graphics: every geometric transformation (rotate, scale, reflect) is a linear map
Signal processing: all linear filters are linear maps on signal vectors
Quantum mechanics: observables are linear operators (linear maps on function spaces)

Learning Resources

Linear transformations

3Blue1Brown

Geometric visualization of linear transformations.

10 min

Quiz

Question 1

The function $T(x_1, x_2) = (x_1 + 1, x_2)$ is a linear map.

Common Mistakes

Thinking translations are linear — they are affine (linear plus a constant offset).
Checking only one of the two linearity conditions — both addition-preservation and scaling-preservation are required.